Assalamu alaikum warahmatullahi wabarakatuhu,
I see a few posts related to ‘rewayat’. I would like to know what this means.
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Basically I'd say this question is to some extent related to Arabic language and also the meaning is related to topics. And therefore it could be considered as too broad.
Nevertheless I think it is reasonably answerable in a general manner due to the fact that even if the topics differ there's a main understanding or agreement.
To make the answer more useful I'll address the topics in which this term actually appears or is mostly noted in the Islamic sciences.
Meaning in the context of Arabic language and the adaptation in the Islamic sciences
In Arabic language a riwayah الرواية (plural riwayat الروايات) refers to a tale, narration, tradition, recital or conversation.
In the Islamic sciences due to the fact that many people were illiterate in the early ages most of the knowledge was spread by oral transmission say: riwaya or riwayah.
Meaning in the context of the qur'an sciences
First note that the statement above is to be taken general: all Islamic sciences have been transmitted orally in first place so the qur'an is orally transmitted nowadays a hafidh who has a high isnad (short narrator chain) will have between 28 and 30 predecessors between him and the prophet ().
Beside this the main topic among the qur'an sciences in which riwaya plays a role or is more correctly speaking is rather noted is the topic of qira'at: in the first three centuries a couple of qira'at have had a high level of acceptance among qur'an students (and later hufadh) and therefore the majority of qurr'a' (recitors) have specialized only on these qira'at and today ten of these qira'at have been considered as the most sane according a couple of rules. Each of these qira'at is what we may call the choices of a qari' and this qar'i has taught his students his choices and traditionally for some reasons each qari' had two rawys who were taught two different choices of recitation.
Today the most prevalent qira'a or better riwayah is the riwaya of Hafs from the qira'a of his teacher 'Asim. This reading was rather fameless until the time of the 'Othman empire (see also Why is the Hafs reading of the Qur'an so prevalent?). While the riwaya of Shu'bah the other student of 'Asim was still more known. Beside the riwaya of Hafs 'an (from) 'Asim today we have the riwaya of Warsh 'an Nafi'i, the rwiya of Qalun 'an Nafi' and the riwaya of as-Sussi 'an abi Amr. Note that in the case of qira'at if a student or rawy had different choices we call these choices tariq الطريق (literally path or road or route) or in plural turuq الطرق for example in the riwaya of Warsh today two main turuq are known that of al-Azraq and that of al-Isbahani.
Note that the term tariq actually is expressive: We have the imam Nafi' who has been taught according to some sources by 70 from among the tabi'is among them 'Abdurrahman ibn Hurmuz al-'A'araj the student of abu Hurraira and abu Ja'afar ibn Yazid al-Qa'aqa'a (one of the ten qurra' and the first imam in the mosque of the prophet in Medina from among the tabi'yn), Shaybah ibn Nisaah, Muslim ibn Jundubal-Hudhali and Yazid ibn Ruman who were students of student of Ubay ibn Ka'ab and Zaid ibn Thabit, some say that some of them even were students of ibn 'Abbas and ibn 'Omar. So from this teacher two students were the most brilliant ones Warsh who came with some prior knowledge from his qur'an teachers in Egypt and wanted to learn the Madani reading and Qalun who was imam Nafi's step son. So this can be considered a first split (into tow paths/road/routes) or a first crossroad as both riwayat even if they have a lot of matches still have differences. Now the second crossroad (leading to different paths) is that of Warsh who taught al-Isbahani (apparently his Madani reading) and al-Azraq (the reading imam Nafi' accepted or corrected from his prior studies) -Note this statement about both turuq is to be taken as an example I'm not sure of its truth-.
Meaning and use in the context of hadith sciences
Therefore for example ahadith are transmitted from a Sahabi who heard or witnessed the prophet() doing (or not doing) or saying (or remaining silence about) something. So the sahabi transmitted it to a tabi'iy (or young sahabi in some cases) and the tabi'i to a tabi'a at-Tabi'i and so on. This tradition of riwaya is still going on from these days. So each narrator who transmitted knowledge of his predecessor (or teacher) to his successor (or student) is called rawy. In hadith sciences scholars of al-Jarh wa T'adil qualify these rawys. An example for a book that still exists these days in different riwayaat as the students who compiled it met the teacher during different time periods where he might have done some corrections is al-Muwatta' of imam Malik today we have around 10 riwayaat of this book at hand from among 100 according to some scholars. The one that is referred to as al-Muwatta' is that of Yahya ibn Yahya al-Laithi who strictly speaking is a "noname" in the hadith sciences he just toke the book and partly dictated it to his son and he just met Malik one year before his death and therefore this is considered the final version of al-Muwatta' therefore in some cases riwayah refers to a version so we say al-Muwatta' in the riwayah (version) of Yahya ibn Yahya al-Laithi or in the riwaah of Muhamad ibn al-Hassan a-Shaybani or al-Qa'anabi etc. Note that even the two sahihs reached us with slightly different versions as at least about imam al-Bukhari i could safely say he began to teach his book when he considered it as finalized so the difference in the wordings comes rather from the listener than from the original.
Also in hadith sciences the terms "tariq" and its plural "turuq" are used however in some cases these turuq may already split much earlier for example the same hadith was reported by different sahabah so the split (into two figurative "paths") is basically at the source of the narrator chain.
Meaning in other Islamic sciences
Here I will just restrict my answer on fiqh:
For example it is well-known that imam a-Shafi'i basically re-edited or adapted his whole views when he moved to Egypt, so anything a student might have taken from his statements while he was in 'Iraq must be taken carefully or considered as his first madhhab. So it is essential to know the timeline when a student met his teacher.
Therefore it often happens that we read in fiqh books that different students reported different statements of their imam (teacher) and all reports of this kind are referred to as riwayat.
Some examples (others may follow later):
So basically in every Islamic knowledge when a students reports from his teacher he is a rawy in some cases there might be opposite riwayat as students may report differently as the teacher might either have changed his mind (in general or for a single case) or/and the students might have met or witnessed their teachers at different time periods.